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Theorem inttrp 25108
Description: The intersection of a non-empty element of a transitive class is a part of the class. (Contributed by FL, 15-Apr-2011.)
Assertion
Ref Expression
inttrp  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A )

Proof of Theorem inttrp
StepHypRef Expression
1 trss 4122 . . . . 5  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
21imp 418 . . . 4  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C_  A )
3 uniss 3848 . . . . 5  |-  ( B 
C_  A  ->  U. B  C_ 
U. A )
4 df-tr 4114 . . . . . . . 8  |-  ( Tr  A  <->  U. A  C_  A
)
5 sstr 3187 . . . . . . . . . . 11  |-  ( ( U. B  C_  U. A  /\  U. A  C_  A
)  ->  U. B  C_  A )
6 intssuni 3884 . . . . . . . . . . . . 13  |-  ( B  =/=  (/)  ->  |^| B  C_  U. B )
7 sstr2 3186 . . . . . . . . . . . . 13  |-  ( |^| B  C_  U. B  -> 
( U. B  C_  A  ->  |^| B  C_  A
) )
86, 7syl 15 . . . . . . . . . . . 12  |-  ( B  =/=  (/)  ->  ( U. B  C_  A  ->  |^| B  C_  A ) )
983ad2ant3 978 . . . . . . . . . . 11  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( U. B  C_  A  ->  |^| B  C_  A ) )
105, 9syl5com 26 . . . . . . . . . 10  |-  ( ( U. B  C_  U. A  /\  U. A  C_  A
)  ->  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A ) )
1110ex 423 . . . . . . . . 9  |-  ( U. B  C_  U. A  -> 
( U. A  C_  A  ->  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A ) ) )
1211com3l 75 . . . . . . . 8  |-  ( U. A  C_  A  ->  (
( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( U. B  C_  U. A  ->  |^| B  C_  A
) ) )
134, 12sylbi 187 . . . . . . 7  |-  ( Tr  A  ->  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( U. B  C_  U. A  ->  |^| B  C_  A )
) )
14133ad2ant1 976 . . . . . 6  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( U. B  C_  U. A  ->  |^| B  C_  A )
) )
1514pm2.43i 43 . . . . 5  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( U. B  C_  U. A  ->  |^| B  C_  A )
)
163, 15syl5com 26 . . . 4  |-  ( B 
C_  A  ->  (
( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A ) )
172, 16syl 15 . . 3  |-  ( ( Tr  A  /\  B  e.  A )  ->  (
( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A ) )
18173adant3 975 . 2  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A ) )
1918pm2.43i 43 1  |-  ( ( Tr  A  /\  B  e.  A  /\  B  =/=  (/) )  ->  |^| B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862   Tr wtr 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-uni 3828  df-int 3863  df-tr 4114
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